Banach–Steinhaus type theorems in locally convex spaces

Banach–Steinhaus type theorems in locally convex spaces

for linear bounded operators

Lahrech Samir ⁱ

Department of mathematics, Mohammed first university, Faculty of sciences, Oujda, Morocco

lahrech@sciences.univ-oujda.ac.ma

Received: 16/3/2003; accepted: 9/6/2004.

Abstract. Banach–Steinhaus type results are established for linear bounded operators be- tween locally convex spaces without barrelledness.

Keywords: Barrelledness, sequential continuity, boundedness, locally convex spaces.

MSC 2000 classification: 47B37 (primary), 46A45.

Introduction

In the past, all of Banach-Steinhaus type results have been established only for some special classes of locally convex spaces, e.g.,barrelled spaces ([2],[3],[4]), s-barrelled spaces ([5]), strictly s-barrelled spaces ([6]), etc. Recently, Cui Chen- gri and Songho Han ([1]) have obtained a Banach-Steinhaus type result which is valid for every locally convex space as follows

1 Theorem. Let (X, λ), (Y, µ) be locally convex spaces and T _n : X → Y bounded linear operators, n ∈ N. If weak-lim

n T _n y = T y exists at each y ∈ X, then the limit operator T send η(X, X ^b )−bounded sets into bounded sets.

In this paper we would like to obtain the same result by taking the topology λ in place of η(X, X ^b ).

Let (X, λ) and (Y, µ) be locally convex spaces. Assume that the locally convex topology µ is generated by the family (q _β ) _β∈I of semi-norms on Y .

An operator T : X → Y is said to be sequentially continuous if {x n } is a sequence in X such that x _n → x then T x n → T x; T is said to be bounded if T sends bounded sets into bounded sets. Clearly, continuous operators are sequentially continuous, and sequentially continuous operators are bounded but in general, converse implications fail. Let X ⁰ , X ^s and X ^b denote the families

i

This work is partially supported by Department of mathematics of Mohammed first uni-

versity in Oujda.

of continuous linear functionals, sequentially continuous linear functionals and bounded linear functionals on X, respectively. In general, the inclusions X ⁰ ⊂ X ^s ⊂ X ^b are strict.

For a linear dual pair (E, F ) let β(E, F ) denote the strongest (E, F ) polar topology on E which is just the topology of uniform convergence on σ(F, E)- bounded subsets of F .

Let C(X λ ), B(X λ ) and C 0 (X _λ ) denote the families of conditionally λ− se- quentially compact sets, bounded sets in (X, λ) and convergent sequences in (X, λ) to 0, respectively.

Let σ ⊂ B(X λ ) such that S

C∈σ

C = X.

Let ζ be the topology on X ^b generated by the family of semi-norms P _C (f ) = sup

y∈C | f(y) |, C ∈ σ.

Let η(X, X _ζ ^s ) denote the topology of uniform convergence on conditionally (X ^s , ζ)− sequentially compacts sets of X ^s .

Remark that η(X, X _ζ ^s ) is coarser than η(X, X ^b ). It follows immediately 2 Proposition. For every locally convex space X the following conditions are equivalent.

(1) For every locally convex space Y and for every sequence {T n } n of bounded linear operators from X into Y such that for every C ∈ σ µ − lim _n T n x = T x uniformly in x ∈ C, the limit operator T is also (λ, µ)-bounded.

(2) (X ^b , ζ) is sequentially complete.

Proof. (1)⇒ (2). Let {f n } be a X ζ ^b − Cauchy sequence in X ^b . then, there exists a linear functional f such that for every C ∈ σ lim _n f _n (x) = f (x) uniformly in x ∈ C. Consequently, f ∈ X ^b by (1).

(2)⇒ (1). Let Y be a locally convex space and {T n } a sequence of bounded linear operators from X into Y such that for every C ∈ σ µ − lim _n T _n x = T x uniformly in x ∈ C. Suppose that B is a bounded subset of X λ and y ⁰ ∈ Y ⁰ . Then there exists β ₀ ∈ I and c 1 > 0 such that for every z ∈ Y | y ⁰ (z) |≤ c 1 q _β

(z).

Therefore, for every C ∈ σ lim _n y ⁰ (T _n x) = y ⁰ (T x) uniformly in x ∈ C. Since y ⁰ ◦ T ⁿ ∈ X ^b for all n ∈ N, y ⁰ ◦ T ∈ X ^b by (2). Therefore, {y ⁰ (T x) : x ∈ B} is bounded. Since y ⁰ ∈ Y ⁰ is arbitrary, T (B) is µ−bounded by the classical Mackey

theorem.

The proof of proposition 1 gives the following

3 Proposition. For every locally convex space X the following conditions are equivalent.

(1) For every locally convex space Y and for every sequence T _n of sequentially

continuous linear operators from X into Y such that for every C ∈ σ µ − lim n T n y = T y uniformly in y ∈ C, the limit operator T is also (λ, µ)-bounded.

(2) X _ζ ^s is sequentially complete.

Assume now that σ satisfies also the condition C(X λ ) ⊂ σ. It follows imme- diately

4 Proposition. X _ζ ^s is sequentially complete.

Proof. Suppose that {A ⁿ } ⁿ is Cauchy sequence in X _ζ ^s . Then,

∀ε > 0 ∀C ∈ σ ∃n 0 ∈ N such that

∀n, m ≥ n 0 ∀y ∈ C | A n y − A m y |< ε. (1)

On the other hand, ∀y ∈ X {A n y} n is Cauchy sequence in R. Consequently, A _n y → Ay in R, as n → ∞. Letting m → ∞ in (1), it follows that

∀ε > 0 ∀C ∈ σ ∃n 0 ∈ N such that

∀n ≥ n 0 ∀y ∈ C | A n y − Ay |≤ ε. (2)

We will show now that A ∈ X ^s .

Let {x n } n ∈ C 0 (X _λ ). Pick any ε > 0. As {x n } ∈ C(X µ ) ⊂ σ, then there exists n ₀ ∈ N such that for all n ∈ N and forall y ∈ C

| A n

x _n − Ax n |≤ ε 2 .

On the other hand, there exists n ₁ ∈ N such that ∀n > n 1 | A n

x _n |≤ ₂ ^ε . In this case

∀n > n 1 | Ax n |≤ ε.

Consequently, A ∈ X ^s . Thus, X _ζ ^s is sequentially complete.

5 Proposition. Let (X, λ), (Y, µ) be locally convex spaces and T n : X → Y sequentially continuous linear operators, n ∈ N. If for every C ∈ σ µ−lim _n T _n z = T z uniformly in z ∈ C, then the limit operator T send η(X, X _ζ ^s )−bounded sets into bounded sets.

Proof. Let y ⁰ ∈ Y ⁰ , C ∈ σ. Then there exists β ⁰ ∈ I and c ¹ > 0 such that for every z ∈ Y | y ⁰ (z) |≤ c 1 q _β

(z). Consequently, sup

z∈C | y ⁰ (T _n z) − y ⁰ (T z) |→ 0.

Since X _ζ ^s is sequentially complete, then {y ⁰ ◦ T n : n ∈ N} is conditionally (X ^s , ζ)−sequentially compact.

Suppose that B is a η(X, X _ζ ^s )−bounded subset of X and {x k } ⊂ B. Then

∃c > 0 ∀k ∈ N ∀n ∈ N | y ⁰ (T _n x _k ) |≤ c. Fix a k ≥ k 0 and ε > 0. Since lim n y ⁰ (T n x _k ) = y ⁰ (T x _k ) there is an n 0 ∈ N such that | y ⁰ (T n

x _k ) − y ⁰ (T x _k ) |< ₂ ^ε . Therefore,

| y ⁰ (T x _k ) |≤| y ⁰ (T x _k ) − y ⁰ (T _n

x _k ) | + | y ⁰ (T _n

x _k ) |< ε 2 + c.

This shows that {y ⁰ (T x) : x ∈ B} is bounded. Since y ⁰ ∈ Y ⁰ is arbitrary, T (B) is µ−bounded by the classical Mackey theorem. Thus, we achieve the

proof.

Let us denote by θ(X, X _ζ ^s ) the topology of uniform convergence on (X ^s , ζ)−

Cauchy sequences. A subset B of X is said to be θ(X, X _ζ ^s )−bounded if for every X _ζ ^s −Cauchy sequence {f n } there exists c > 0 such that for every sequence {x k } in B | f n (x _k ) |≤ c ∀n ∈ N ∀k ∈ N.

Then the proof of proposition 4 gives the following.

6 Proposition. Let (X, λ), (Y, µ) be locally convex spaces and T _n : X → Y bounded linear operators, n ∈ N. If for every C ∈ σ µ − lim _n T _n y = T y uniformly in y ∈ C, then the limit operator T send θ(X, X _ζ ^s )bounded sets into bounded sets.

Now we have a useful proposition as follows.

7 Proposition. Let (X, λ), (Y, µ) be locally convex spaces and T _n : X → Y sequentially continuous linear operators, n ∈ N. If for every C ∈ σ µ−lim _n T n y = T y uniformly in y ∈ C, then the limit operator T send λ−bounded sets into bounded sets.

Proof. By propositions 2 and 3, we deduce the result.

References

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